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NUMBERS in VS .NET
NUMBERS QR Code JIS X 0510 Scanner In VS .NET Using Barcode Control SDK for VS .NET Control to generate, create, read, scan barcode image in .NET applications. Making QR Code 2d Barcode In .NET Using Barcode maker for .NET framework Control to generate, create QR Code JIS X 0510 image in Visual Studio .NET applications. [CHAP. 1
Quick Response Code Recognizer In .NET Using Barcode recognizer for .NET Control to read, scan read, scan image in Visual Studio .NET applications. Bar Code Creation In .NET Using Barcode creator for .NET Control to generate, create bar code image in .NET applications. 1.107. (a) If x is a rational number whose square is less than 2, show that x 2 x2 =10 is a larger such number. (b) If x is a rational number whose square is greater than 2, nd in terms of x a smaller rational number whose square is greater than 2. 1.108. Illustrate how you would use Dedekind cuts to de nep p p p p p p p (a) 5 3; b 3 2; c 3 2 ; d 2= 3. Barcode Recognizer In .NET Framework Using Barcode reader for Visual Studio .NET Control to read, scan read, scan image in .NET framework applications. QR Code JIS X 0510 Encoder In Visual C# Using Barcode printer for Visual Studio .NET Control to generate, create QR image in .NET applications. Sequences
QR Creation In VS .NET Using Barcode drawer for ASP.NET Control to generate, create QR Code ISO/IEC18004 image in ASP.NET applications. Creating Quick Response Code In Visual Basic .NET Using Barcode printer for .NET Control to generate, create QR image in Visual Studio .NET applications. DEFINITION OF A SEQUENCE A sequence is a set of numbers u1 ; u2 ; u3 ; . . . in a de nite order of arrangement (i.e., a correspondence with the natural numbers) and formed according to a de nite rule. Each number in the sequence is called a term; un is called the nth term. The sequence is called nite or in nite according as there are or are not a nite number of terms. The sequence u1 ; u2 ; u3 ; . . . is also designated brie y by fun g. UPC A Maker In .NET Using Barcode maker for VS .NET Control to generate, create UPC Symbol image in .NET applications. Bar Code Generator In VS .NET Using Barcode printer for Visual Studio .NET Control to generate, create barcode image in .NET framework applications. EXAMPLES. 1. The set of numbers 2; 7; 12; 17; . . . ; 32 is a nite sequence; the nth term is given by un 2 5 n 1 5n 3, n 1; 2; . . . ; 7. 2. The set of numbers 1; 1=3; 1=5; 1=7; . . . is an in nite sequence with nth term un 1= 2n 1 , n 1; 2; 3; . . . . USS Code 39 Generation In VS .NET Using Barcode drawer for .NET Control to generate, create Code 3 of 9 image in VS .NET applications. Making UPC  E1 In .NET Framework Using Barcode creation for VS .NET Control to generate, create UPC E image in .NET applications. Unless otherwise speci ed, we shall consider in nite sequences only.
EAN / UCC  13 Scanner In VS .NET Using Barcode decoder for Visual Studio .NET Control to read, scan read, scan image in VS .NET applications. Bar Code Scanner In .NET Framework Using Barcode scanner for .NET framework Control to read, scan read, scan image in VS .NET applications. LIMIT OF A SEQUENCE A number l is called the limit of an in nite sequence u1 ; u2 ; u3 ; . . . if for any positive number we can nd a positive number N depending on such that jun lj < for all integers n > N. In such case we write lim un l. Encode UCC128 In Java Using Barcode printer for BIRT reports Control to generate, create EAN / UCC  13 image in BIRT reports applications. Matrix 2D Barcode Generation In Java Using Barcode generation for Java Control to generate, create 2D Barcode image in Java applications. EXAMPLE . If un 3 1=n 3n 1 =n, the sequence is 4; 7=2; 10=3; . . . and we can show that lim un 3. Bar Code Drawer In None Using Barcode maker for Software Control to generate, create barcode image in Software applications. Make UPCA Supplement 2 In Java Using Barcode creation for Android Control to generate, create UPCA image in Android applications. If the limit of a sequence exists, the sequence is called convergent; otherwise, it is called divergent. A sequence can converge to only one limit, i.e., if a limit exists, it is unique. See Problem 2.8. A more intuitive but unrigorous way of expressing this concept of limit is to say that a sequence u1 ; u2 ; u3 ; . . . has a limit l if the successive terms get closer and closer to l. This is often used to provide a guess as to the value of the limit, after which the de nition is applied to see if the guess is really correct. Code128 Printer In Java Using Barcode creator for Android Control to generate, create Code128 image in Android applications. ANSI/AIM Code 128 Recognizer In None Using Barcode decoder for Software Control to read, scan read, scan image in Software applications. THEOREMS ON LIMITS OF SEQUENCES If lim an A and lim bn B, then
n!1 n!1 1. 2. 3. n!1 n!1 n!1 lim an bn lim an lim bn A B
n!1 n!1 n!1 n!1 lim an bn lim an lim bn A B lim an bn lim an lim bn AB
n!1 n!1 Copyright 2002, 1963 by The McGrawHill Companies, Inc. Click Here for Terms of Use.
24 lim an n!1 an A n!1 bn lim bn B lim
SEQUENCES
[CHAP. 2
if lim bn B 6 0
a If B 0 and A 6 0, lim n does not exist. n!1 bn a If B 0 and A 0, lim n may or may not exist. n!1 bn 5. 6. n!1 p lim an lim an p A p , n!1 for p any real number if A p exists.
lim pan p n!1 pA , liman
for p any real number if pA exists.
INFINITY We write lim an 1 if for each positive number M we can nd a positive number N (depending on n!1 M) such that an > M for all n > N. Similarly, we write lim an 1 if for each positive number M we can nd a positive number N such that an < M for all n > N. It should be emphasized that 1 and 1 are not numbers and the sequences are not convergent. The terminology employed merely indicates that the sequences diverge in a certain manner. That is, no matter how large a number in absolute value that one chooses there is an n such that the absolute value of an is greater than that quantity. BOUNDED, MONOTONIC SEQUENCES If un @ M for n 1; 2; 3; . . . ; where M is a constant (independent of n), we say that the sequence fun g is bounded above and M is called an upper bound. If un A m, the sequence is bounded below and m is called a lower bound. If m @ un @ M the sequence is called bounded. Often this is indicated by jun j @ P. Every convergent sequence is bounded, but the converse is not necessarily true. If un 1 A un the sequence is called monotonic increasing; if un 1 > un it is called strictly increasing. Similarly, if un 1 @ un the sequence is called monotonic decreasing, while if un 1 < un it is strictly decreasing. EXAMPLES. 1. The sequence 1; 1:1; 1:11; 1:111; . . . is bounded and monotonic increasing. It is also strictly increasing. 2. The sequence 1; 1; 1; 1; 1; . . . is bounded but not monotonic increasing or decreasing. 3. The sequence 1; 1:5; 2; 2:5; 3; . . . is monotonic decreasing and not bounded. However, it is bounded above. The following theorem is fundamental and is related to the Bolzano Weierstrass theorem ( 1, Page 6) which is proved in Problem 2.23. Theorem. Every bounded monotonic (increasing or decreasing) sequence has a limit. LEAST UPPER BOUND AND GREATEST LOWER BOUND OF A SEQUENCE A number M is called the least upper bound (l.u.b.) of the sequence fun g if un @ M, n 1; 2; 3; . . . while at least one term is greater than M for any > 0. " " A number m is called the greatest lower bound (g.l.b.) of the sequence fun g if un A m, n 1; 2; 3; . . . " while at least one term is less than m for any > 0. Compare with the de nition of l.u.b. and g.l.b. for sets of numbers in general (see Page 6). CHAP. 2]

